Subgroups — C₃×#10604098 #14312643

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65718	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×#64844	C₂	1	2	yes	maximal, complemented
5	C₃×C₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	#10604098	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4130	C₄	1	1	yes	characteristic
8	C₃×D₅⋊M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₃×#3998	C₄	1	2	yes	complemented
11	C₃×#3992	C₄	1	2	yes	—
12	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
13	C₃×#3990	C₄	1	2	yes	complemented
14	C₃×D₅⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×#672	C₄	1	1	yes	characteristic, complemented
17	C₃×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
18	C₃×D₅×C₁₆	C₄	1	1	yes	characteristic, complemented
19	C₃×D₅×C₁₆	C₂²	1	1	yes	characteristic
20	C₃×D₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₃×#2108	—	5	1	no	maximal, Hall
22	#65718	C₂×C₃	1	1	yes	characteristic
23	D₅⋊M₃₂	C₂×C₃	1	1	yes	characteristic
24	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic
25	#64844	C₂×C₃	1	2	yes	complemented
26	C₁₆×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
27	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₂²⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
29	C₃×#704	C₈	1	2	yes	—
30	C₃×C₄⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
31	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
32	C₃×#700	C₈	1	2	yes	—
33	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
40	C₃×#674	C₂×C₄	1	4	yes	—
41	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₃×#672	C₂×C₄	1	1	yes	characteristic
44	C₃×#672	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×M₃₂	—	5	1	no	—
46	C₂²×C₃×C₁₆	—	5	1	no	—
47	C₃×#405	—	5	1	no	—
48	C₃×#346	—	5	2	no	—
49	C₃×C₄×C₁₆	—	5	2	no	—
50	#4130	C₃×C₄	1	1	yes	characteristic
51	D₅⋊M₁₆	C₃×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
53	#3998	C₃×C₄	1	2	yes	complemented
54	#3992	C₃×C₄	1	2	yes	—
55	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
56	#3990	C₃×C₄	1	2	yes	complemented
57	D₅⋊C₁₆	C₂²×C₃	1	2	yes	—
58	C₂×C₅×C₁₆	C₃×C₄	1	1	yes	characteristic, complemented
59	C₂×#672	C₃×C₄	1	1	yes	characteristic, complemented
60	D₅×C₁₆	C₃×C₄	1	1	yes	characteristic, complemented
61	D₅×C₁₆	C₃×C₄	1	1	yes	characteristic, complemented
62	D₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
63	D₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
64	#2108	—	5	1	no	Sylow
65	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
69	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
71	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
73	C₃×#170	C₂×C₈	1	4	yes	—
74	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×#168	C₄²	1	1	yes	characteristic
77	C₃×#168	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×M₁₆	—	5	1	no	—
79	C₂²×C₃×C₈	—	5	1	no	—
80	C₃×#117	—	5	1	no	—
81	C₃×M₃₂	—	5	2	no	—
82	C₃×M₃₂	—	5	2	no	—
83	C₂×C₃×C₁₆	—	5	2	no	—
84	C₂×C₃×C₁₆	—	5	1	no	—
85	C₂×C₃×C₁₆	—	5	1	no	—
86	C₂×C₃×C₁₆	—	5	1	no	—
87	C₂×C₃×C₁₆	—	5	1	no	—
88	C₂×C₃×C₁₆	—	5	1	no	—
89	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
90	C₃×#97	—	5	2	no	—
91	C₃×C₄×C₈	—	5	2	no	—
92	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂²⋊AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
94	#704	C₃×C₈	1	2	yes	—
95	C₄⋊AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
96	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
97	#700	C₃×C₈	1	2	yes	—
98	D₅⋊C₈	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
101	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
105	#674	C₂×C₃×C₄	1	4	yes	—
106	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
107	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
108	#672	C₂×C₃×C₄	1	1	yes	characteristic
109	#672	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×M₃₂	—	5	1	no	—
111	C₂²×C₁₆	—	5	1	no	—
112	#405	—	5	1	no	—
113	#346	—	5	2	no	—
114	C₄×C₁₆	—	5	2	no	—
115	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
118	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
119	C₃×AGL(1,𝔽₅)	—	2	4	no	—
120	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
123	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₃×M₁₆	—	5	2	no	—
126	C₃×M₁₆	—	5	2	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₂×C₃×C₈	—	5	1	no	—
129	C₂×C₃×C₈	—	5	1	no	—
130	C₂×C₃×C₈	—	5	1	no	—
131	C₂×C₃×C₈	—	5	1	no	—
132	C₂×C₃×C₈	—	5	1	no	—
133	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	5	2	no	—
135	C₃×C₂²⋊C₄	—	5	2	no	—
136	C₃×C₄²	—	5	2	no	—
137	C₃×C₁₆	—	5	4	no	—
138	C₃×C₁₆	—	5	1	no	—
139	C₃×C₁₆	—	5	1	no	—
140	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
143	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	4	yes	—
144	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
145	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
146	C₄×D₅	C₃×C₄²	1	1	yes	characteristic
147	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
148	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
149	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
150	#170	C₂×C₃×C₈	1	4	yes	—
151	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
152	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
153	#168	C₃×C₄²	1	1	yes	characteristic
154	#168	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×M₁₆	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	#117	—	5	1	no	—
158	M₃₂	—	5	2	no	—
159	M₃₂	—	5	2	no	—
160	C₂×C₁₆	—	5	2	no	—
161	C₂×C₁₆	—	5	1	no	—
162	C₂×C₁₆	—	5	1	no	—
163	C₂×C₁₆	—	5	1	no	—
164	C₂×C₁₆	—	5	1	no	—
165	C₂×C₁₆	—	5	1	no	—
166	C₂×C₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
167	#97	—	5	2	no	—
168	C₄×C₈	—	5	2	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
171	C₃×D₅	—	2	1	no	—
172	C₃×D₅	#504	1	2	yes	—
173	C₂³×C₃	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	4	no	—
180	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₃×C₈	—	5	4	no	—
182	C₃×C₈	—	5	1	no	—
183	C₃×C₈	—	5	1	no	—
184	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	C₃×C₄×C₈	1	1	yes	characteristic
189	C₂×D₅	C₂²×C₃×C₈	1	1	yes	characteristic
190	AGL(1,𝔽₅)	—	2	4	no	—
191	C₄×C₅	C₃×C₄×C₈	1	1	yes	characteristic
192	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
193	Dic₅	C₃×C₄×C₈	1	1	yes	characteristic
194	Dic₅	C₂²×C₃×C₈	1	1	yes	characteristic
195	C₂²×C₄	—	5	1	no	—
196	M₁₆	—	5	2	no	—
197	M₁₆	—	5	2	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	5	1	no	—
202	C₂×C₈	—	5	1	no	—
203	C₂×C₈	—	5	1	no	—
204	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₄⋊C₄	—	5	2	no	—
206	C₂²⋊C₄	—	5	2	no	—
207	C₄²	—	5	2	no	—
208	C₁₆	—	5	4	no	—
209	C₁₆	—	5	1	no	—
210	C₁₆	—	5	1	no	—
211	C₁₆	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₁₆	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	—	5	1	no	—
215	C₂²×C₃	—	5	1	no	—
216	C₂²×C₃	—	10	2	no	—
217	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₃×C₄	—	5	1	no	—
219	C₃×C₄	—	5	1	no	—
220	C₃×C₄	—	10	4	no	—
221	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
225	D₅	—	2	1	no	—
226	D₅	C₃×#504	1	2	yes	—
227	C₂³	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	4	no	—
234	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₈	—	5	4	no	—
236	C₈	—	5	1	no	—
237	C₈	—	5	1	no	—
238	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	5	2	no	—
242	C₂×C₃	—	10	1	no	—
243	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₅	C₃×#2108	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	5	1	no	—
246	C₂²	—	5	1	no	—
247	C₂²	—	10	2	no	—
248	C₂²	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₄	—	5	1	no	—
250	C₄	—	5	1	no	—
251	C₄	—	10	4	no	—
252	C₄	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃	#10604098	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	5	2	no	—
257	C₂	—	10	1	no	—
258	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₁₆	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₈	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂×C₃×C₅	C₂×C₄×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₈×D₅	C₂³×C₃
hypercenter	C₂×C₃×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₁₆ ◁ C₂×C₃×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₃×#64844	C₂	semidirect
5. C₃×C₁₆×AGL(1,𝔽₅)	C₂	semidirect
6. #10604098	C₃	direct (N × H)
10. C₃×#3998	C₄	semidirect
13. C₃×#3990	C₄	semidirect
15. C₂×C₃×C₅×C₁₆	C₄	semidirect
16. C₂×C₃×#672	C₄	semidirect
17. C₃×D₅×C₁₆	C₄	semidirect
18. C₃×D₅×C₁₆	C₄	semidirect
25. #64844	C₂×C₃	semidirect
26. C₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
53. #3998	C₃×C₄	semidirect
56. #3990	C₃×C₄	semidirect
58. C₂×C₅×C₁₆	C₃×C₄	semidirect
59. C₂×#672	C₃×C₄	semidirect
60. D₅×C₁₆	C₃×C₄	semidirect
61. D₅×C₁₆	C₃×C₄	semidirect
89. C₂×C₃×C₁₆	AGL(1,𝔽₅)	semidirect
166. C₂×C₁₆	C₃×AGL(1,𝔽₅)	semidirect
213. C₃×C₅	#2108	semidirect
244. C₅	C₃×#2108	semidirect
254. C₃	#10604098	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#65718	2
C₃×D₅⋊M₃₂	2
C₂×C₃×D₅×C₁₆	2
C₃×#64844	2
C₃×C₁₆×AGL(1,𝔽₅)	2
#10604098	3
C₃×#2108	5

Subgroups — C3×#10604098 #14312643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604098 #14312643